Let $M$ be a $4n$-dimensional spin manifold with a fixed Riemannian metric $g$. Let $S$ be a spinor bundle over $M$ and fix the Riemannian connection on it. There is a decomposition $S=S^+\oplus S^-$, where $S^\pm$ is the $\pm$1-eigenbundle with respect to the Clifford multiplication by $\omega=e_1\cdot ...\cdot e_{4n}$, where $\{e_1,...,e_{4n}\}$ is a positively oriented orthonormal basis of $T_xM$ at every $x\in M$.

Now let $t:M\rightarrow M$ be an orientation and spin structure preserving involution which is an isometry w.r.t. the metric $g$. This involution lifts to an action $T:S\rightarrow S$ on the spinor bundle.

How does one show that $T$ preserves the above $\mathbb{Z}_2$-grading and that it commutes with the Dirac operator $D:\Gamma(S)\rightarrow \Gamma(S)$ defined by $D\sigma=\sum e_j\cdot \nabla_{e_j}\sigma$, where $\nabla$ is the covariant derivative associated to the Riemannian connection?